Is there any method to solve (find the number of solutions or in some problems find the solutions) problems of following type:
1.${}\quad x=\log_e x^2-1$
2.${}\quad \tan x+2x^2-3=0$
other than by separating polynomial function and other types of functions (like exponential, logarithmic) writing them on two sides of equality and drawing their graphs and noting where the two curves intersect?
Any sort of approach or method is welcome as long as it leads to an elegant solution and as is elementary.